Given a short exact sequence of modules
(1)
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let
(2)
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(3)
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be projective resolutions of and , respectively. Then there is a projective resolution of
(4)
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such that the above diagrams are commutative. Here, is the injection of the first summand, whereas is the projection onto the second factor for .
The name of this lemma derives from the shape of the diagram formed by the short exact sequence and the given projective resolutions.